The most promising candidates are topological quantum systems which would protect information encoded in their degenerate groundspace while interacting with a thermal environment. Many models have been suggested but several approaches have been shown to fail due to no-go results of increasingly general scope. In a nutshell, 2D topological models and many 3D topological models have point-like excitations which propagate freely and change the groundstate at any non-zero temperature. A recent suggestion is to introduce effective long-range interactions between those point-like excitations.
In this presentation, I will first explain the desiderata for self-correction, review the recent advances and no-go results, and describe the current endeavours to define a self-correcting system in 2D and 3D. Time permitting, I will briefly present our recent work on the thermal instability of models which aim to introduce effective long-range interactions between point-like excitations (joint work with Beni Yoshida, John Preskill and David Poulin).